Thursday, November 3, 2016 - 3:15pm - 4:05pm
Svetlana Roudenko (George Washington University)
We discuss the focusing nonlinear Klein-Gordon equation starting with the cubic nonlinearity in 3 dimensions. Inspired by the paper of Donninger-Schlag on this equation, we further investigate the blow up and scattering behavior of its solutions. We extend the theoretical boundaries of the blow up regions and discuss the behavior of solutions there, for example, formation of a singularity away from the origin, and behavior near the ground and excited states. We also show extensions to other dimensions and nonlinearities.
Tuesday, November 1, 2016 - 3:15pm - 4:05pm
Justin Holmer (Brown University)
We consider a version of the nonlinear Schroedinger equation (NLS) with point nonlinearity, which can formulated as the linear Schroedinger equation away from the spatial origin together with a nonlinear jump condition in the derivative across the origin. This model can be viewed as a limiting form of a concentrated nonlinearity and exhibits many of the same properties as the standard nonlinear Schroedinger equation. In fact, in most cases, the analysis is simpler than for standard NLS and thus simpler proofs and/or stronger results are possible.
Tuesday, November 1, 2016 - 11:30am - 12:20pm
Christof Sparber (University of Illinois, Chicago)
The possibility of finite-time, dispersive blow up for nonlinear equations of Schrödinger type is revisited. We extend earlier results in the literature to include the multi-dimensional case, as well as the case of Davey-Stewartson and Gross-Pitaevskii equations. As a by-product of our analysis, a sharp global smoothing estimate for the integral term appearing in Duhamel’s formula is obtained.
Thursday, July 30, 2009 - 11:00am - 11:50am
Alexis Vasseur (The University of Texas at Austin)
In this talk, we present the study of the regularity of solutions to some
systems of reaction–diffusion equations, with reaction terms having a
subquadratic growth. We show the global boundedness and regularity of
solutions, without smallness assumptions, in any dimension N. The proof
is based on blow-up techniques. The natural entropy of the system plays a
crucial role in the analysis. It allows us to use of De Giorgi type
methods introduced for elliptic regularity with rough coefficients. Even
Monday, June 3, 2013 - 9:00am - 9:40am
Bjorn Sandstede (Brown University)
Many planar spatially extended systems exhibit localized standing structures such as pulses and oscillons. In particular, such structures can emerge at Turing and forced Hopf bifurcations. In this talk, I will give an overview of these mechanisms and show how geometric blow-up techniques can be used to analyze them: among the findings is the bifurcation of localized structures that have significantly larger amplitudes than expected from formal considerations. This is joint work with Kelly McQuighan, David Lloyd, and Scott McCalla.
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